= BCS pairing-field effective action
{c}
{title2=$S_{\rm eff}[\Delta]=\int|\Delta|^2/\lambda-\operatorname{Tr}\ln\mathcal G^{-1}[\Delta]$}
An attractive local pairing interaction can be decoupled with a complex <Hubbard–Stratonovich transformation>. Integrating the fermionic <Grassmann variables> gives an <effective action> for the pair field. Its uniform saddle is the <BCS gap equation>, and its slow-field expansion near the transition gives the <Ginzburg-Landau theory of superconductivity>. A <trace> over a fully doubled particle-hole basis requires the corresponding factor of one half; the displayed formula uses one independent pairing block.
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